- Online publication date: March 2017

- Publisher: Cambridge University Press
- DOI: https://doi.org/10.1017/9781316755822.007
- pp 123-159

Abstract. The relation between intensionality as understood in philosophy and in metamathematics (in particular in early writings of S. Feferman) is explored in this paper. It investigates whether the latter can be interpreted as an instance of the former and presents metamathematical examples of “serious intensionality”.

Truth and reference (i.e., extensions) on the one side and meaning (i.e., intensions) on the other side are closely related: for example, principles like every true sentence is meaningful and expressions with the same meaning refer to the same things (if they refer) seem to be correct from an intuitive point of view. Of course, this does not mean that, e.g., truth or falsity are the meanings of (declarative) sentences. Nonetheless, were it not because of contexts taken from ordinary language — like those containing modalities and propositional attitudes — expressions like “meaning” could perhaps be explained by employing terminology taken solely from referential semantics. Thus, it has been claimed that for scientific purposes one may well get on with purely extensional languages.1 In fact, this seems to be “obviously” true for the mathematical discourse.

It is mainly due to the work of S. Feferman that the topic of intensionality has nevertheless gained some relevance, perhaps even popularity, in the field of metamathematics (cf. primarily [12]). But what is the relation between “intensionality” as understood in philosophy and as understood in metamathematics? Are these two concepts actually the same? If not, is it at least possible to make them fruitful for each other, e.g., by construing one of them as a special case of the other? This paper addresses these questions both from a conceptual perspective and by presenting relevant metamathematical results.

Intensionality in philosophy: an overview.

Intensionality: basic phrases. When dealing with intensionality, a natural starting point is G. Frege's distinction (see his seminal [16]) between “Bedeutung” (i.e., reference) and “Sinn” (i.e., meaning): expressions s, t may refer to the same entity while, at the same time, presenting that entity in different ways — whence s and t have different meanings.

[1] Situations and attitudes, MIT Press, Cambridge,Mass., 1983. and ,

[2] Foundations of constructive mathematics, Springer, New York, 1985. ,

[3] The logic of provability, Cambridge UP, Cambridge, 1993. ,

[4] Chance, cause, reason, The University of Chicago Press, Chicago, London, 1963. ,

[5] First-order proof theory of arithmetic, Handbook of proof theory ( , editor), Elsevier, Amsterdam, 1998, pp. 79–147. ,

[6] 1928, Quotations are from The logical structure of the world, University of California Press, Berkeley, 1967. , Der logische Aufbau der Welt, ,

[7] Logische Syntax der Sprache, Springer, Wien, 1934, Quotations are from The logical syntax of language, Routledge and Kegan Paul, 1937. ,

[8] Meaning and necessity, University of Chicago Press, Chicago, 1947. ,

[9] Structured meanings, MIT Press, Cambridge,Mass., 1985. ,

[10] Inquiries into truth and interpretation, Clarendon Press, Oxford, 1984. ,

[11] Hilbert's program, Reidel, Dordrecht, 1986. ,

[12] Arithmetization of metamathematics in a general setting, FundamentaMathematicae, vol. XLIX (1960), pp. 35–92. ,

[13] Logic, methodology and philosophy of science III ( and , editors), North-Holland, Amsterdam, 1968, pp. 121–135. , Autonomous transfinite progressions and the extent of predicative mathematics,

[14] Constructive theories of functions and classes, Logic colloquium '78 ( et al., editors), Springer, Berlin, 1979, pp. 159–224. ,

[15] Intensionality in mathematics, The Journal of Symbolic Logic, vol. 14 (1985), pp. 41–55. ,

[16] Zeitschrift fÜr Philosophie und philosophische Kritik, NF, vol. 100 (1892), pp. 25–50. Reprinted in (ed.): G. Frege: Funktion, Begriff, Bedeutung, Fünf logische Studien, Vandenhoeck and Ruprecht, Göttingen 1962, 40–65. , Ü ber Sinn und Bedeutung,

[17] Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisse eines mathematischen Kolloquiums, vol. 4 (1933), pp. 39–40. ,

[18] Rosser sentences, Annals ofMathematical Logic, vol. 16 (1979), pp. 81–99. and ,

[19] Metamathematics of first-order arithmetic, Springer, Berlin, 1993. and ,

[20] Mathematics without numbers, Clarendon, Oxford, 1989. ,

[21] A companion to modal logic, Methuen, London/New York, 1984. and ,

[22] Models of peano arithmetic, Clarendon Press, Oxford, 1991. ,

[23] Semantical considerations on modal logic, Acta Philosophica Fennica, vol. 16 (1963), pp. 83–94. ,

[24] Set theory, Elsevier, Amsterdam, 1980. ,

[25] Symbolic logic, Century, New York, 1932. and ,

[26] Extensionality, Mind, vol. 69 (1960), pp. 55–62. ,

[27] Logic colloquium '90 ( and , editors), Springer, Berlin, 1993. , Sense and denotation as algorithm an value,

[28] Thirty years of foundational studies, Blackwell, Oxford, 1966. ,

[29] , “Natural” representations and extensions of Gödel's second theorem, forthcoming.

[30] Zur Metamathematik nichtaxiomatisierbarer Theorien, CIS,München, 1996. ,

[31] Argument und Analyse. proceedings of GAP4 ( and , editors), mentis, Paderborn, 2002, pp. 109–136. , On the limits of Gödel's second incompleteness theorem,

[32] Hilbert's programme and Gödel's theorems, Dialectica, vol. 56 (2002), pp. 347–370. and ,

[33] Mathematics without foundations, Journal of Philosophy, vol. 64 (1967), pp. 5–22. ,

[34] Notes on existence and necessity, Journal of Philosophy, vol. 40 (1943), pp. 113–127. ,

[35] Mathematical logic, revised ed., Harvard University Press, Cambridge, 1951. ,

[36] From a logical point of view ( , editor), Harvard University Press, Cambridge,Mass., 1953, pp. 47–64. , The problem of meaning in linguistics,

[37] From a logical point of view ( , editor), Harvard University Press, Cambridge,Mass., 1953, pp. 139–159. , Reference and modality,

[38] Word and object, M.I.T. Press, Cambridge,Mass., 1960. ,

[39] Ontological relativity and other essays, Columbia University Press, New York, 1969. , Ontological relativity,

[40] An inquiry into meaning and truth, Unwin Paperbacks, London, 1950. ,

[41] Intensional mathematics, North-Holland, Amsterdam, 1985. (editor),

[42] Consistency and related metamathematical properties, Technical Report 75–02, Mathematisch Instituut, Amsterdam, 1975. ,

[43] Handbook of mathematical logic ( , editor), North-Holland, Amsterdam, 1977. , The incompleteness theorems,

[44] Self-reference and modal logic, Springer, Berlin, 1985. ,

[45] Peano's smart children: a provability logical study of systems with built-in consistency, Notre Dame Journal of Formal Logic, vol. 30 (1989), pp. 161–196. ,